Harmony is the underlying foundation of Western art music. Harmony is the study of how particular sonorities are related and function with respect to a primary tonal region based upon a central pitch class.
The Mathematical definition of harmony: Presume that waves X and Y are of wavelengths A and B. In other words X(nA) = X(mA), and Y(nB) = Y(mB), where n and m are any 2 integer numbers.
Now, presume that wave Z is a combination of X and Y. In other words Z(n) = X(n) + Y(n), where n is any number. Because Z is a function of X+Y, Z only repeats at any point where X and Y both repeat. This point is the lowest common multiple of A and B, which will henceforth be referred to as value C. C is the combined wavelength of X and Y.
The nature of tonal harmony is that the lower the value of C, the more harmonized the notes are. This is why octaves are the most harmonized; a note's octave repeats twice as often as the note itself, so if 2n is the wavelength of any note, n is its octave. Obviously, the lowest common multiple of N and 2N is 2N itself, which makes the wavelength of a note combined with its octave just the wavelength of the note itself, and the shortest possible combine wavelength of 2 notes.